[Linear Algebra] Linear Combination, Vector Equation, Span

Linear Combinations

Given vectors $\textbf{v}_1, \textbf{v}_2, \cdots, \textbf{v}_p$ in $\mathbb{R}^{n}$ and given scalars $c_1, c_2, \cdots, c_p$

$c_1 \textbf{v}_1 + \cdots + c_p \textbf{v}_p$

is called a linear combination of $\textbf{v}_1, \cdots, \textbf{v}_p$ with weights or coefficients $c_1, c_2, \cdots, c_p$

The weights in a linear combination can be any real numbers, including zero

From Matrix Equation to Vector Equation

• A matrix equation can be converted into a vector equation

e.g.
$2 x_1 + 2 x_2 + 1 x_3 = 6$
$3 x_1 + 1 x_2 + 0 x_3 = 14$
$1 x_1 + 3 x_2 + 1 x_3 = 18$

$\begin{bmatrix} 2 & 2 & 1 \\ 3 & 1 & 0 \\ 1 & 3 & 1 \\ \end{bmatrix}\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} = \begin{bmatrix} 6 \\ 14 \\ 18 \end{bmatrix}$

$\begin{bmatrix} 2\\ 3\\ 1\end{bmatrix} x_1 + \begin{bmatrix} 2\\ 1\\ 3\end{bmatrix} x_2 + \begin{bmatrix} 1\\ 0\\ 1\end{bmatrix} x_3 = \begin{bmatrix} 6\\ 14\\ 18\end{bmatrix}$

Span

Definition : given a set of vectors $\textbf{v}_1, \cdots, \textbf{v}_p \in \mathbb{R}^{n}$, $\textrm{Span} \begin{Bmatrix} \textbf{v}_1, \cdots, \textbf{v}_p \end{Bmatrix}$ is defined as the set of all linear combinations of $\textbf{v}_1, \cdots, \textbf{v}_p$

$\textrm{Span} \begin{Bmatrix} \textbf{v}_1, \cdots, \textbf{v}_p \end{Bmatrix} = c_1 \textbf{v}_1 + c_2 \textbf{v}_2 + \cdots c_p \textbf{v}_p$ with arbitrary scalars $c_1, \cdots, c_p$

$\textrm{Span} \begin{Bmatrix} \textbf{v}_1, \cdots, \textbf{v}_p \end{Bmatrix}$ is also called the subset of $\mathbb{R}^{n}$ spanned by $\textbf{v}_1, \cdots, \textbf{v}_p$